Hi folks! I just stumbled onto this forum... wandering far from my normal stomping grounds in the Edibles board. I have some succulents (including a holiday cactus of some sort) that I'd love help IDing, but that's it's own thread.
But I saw your question about symmetry, JB, and figured I could give you a hand there. The easiest way for me to visualize it is to think of holding a mirror up to the bloom (no need for a real mirror, just a good imagination). Bilateral (literally "two sides") symmetry will only let you hold the mirror in one direction to get the same picture on both sides. People are bilaterally symmetrical - if you put a line up and down through the nose and navel, we're the same on both sides, but if you put it crossways like a belt, we aren't.
This flower is bilaterally symmetrical:
You have to hold the mirror vertically, because the petals are different sizes on the top and bottom (from this angle of view).
Radial (from the same root as radius, the measure of a circle) symmetry is like a wheel. You can put your mirror at any angle, as long as it passes through the middle point, and you'll see the same thing on both sides.
This flower is radially symmetrical:
The petals are the same size all the way around, so no matter how you hold your mirror, you get two matching halves.
[Both photos from Cahac, posted in this thread. I hope you don't mind me using them as examples.]